CANADIAN BOARD OF EXAMINERS FOR PROFESSIONAL SURVEYORS ATLANTIC PROVINCES BOARD OF EXAMINERS FOR LAND SURVEYORS ________________________________________________________________________
SCHEDULE I / ITEM 2 March 1007
LEAST SQUARES ESTIMATION AND DATA ANALYSIS
Note: This examination consists of 8 questions on 3 pages. Marks
Q. No Time: 3 hours Value Earned
1
Given the following mathematical models
2 2
1 1
0 ) ,
(
0 ) , (
2 1 2 2
1 1 1
x x
C C x
, x f
C C x
f
l l
l l
=
=
=
=
=
=
=
=
where f1andf2 are vectors of mathematical models, x1and x2 are vectors of unknown parameters, l1andl2 are vectors of observations,
2 1
2
1 ,C ,C and C
Cl l x x are covariance matrices.
a) Linearize the mathematical models b) Formulate the variation function
10
2
Given n independent measurements of a single quantity: l1,l2,...,ln and their corresponding standard deviations:σ1,σ2,...,σn, derive the expression for the variance of the mean value using the Law of
Propagation of Variances. 10
3
Given the following variance-covariance matrix from a least squares adjustment for the coordinates of point A and B,
2
x m
5 2 1 1
2 5 1 2
1 1 10 5
1 2 5 10 C
= where x=[xA yA xB yB]T.
a) Compute the correlation coefficient between xA and yB. b) Compute the correlation coefficient between yA and xB.
c) Compute the variance-covariance matrix for the coordinate differences
A B x x
x= −
∆ and ∆y=yB −yA.
10
4
Given the covariance matrix of the horizontal coordinates (x, y) of a survey station, determine the semi-major, semi-minor axis and the orientation of the standard error ellipse associated with this station.
x m2 0196 . 0 0246 . 0
0246 . 0 0484 .
C 0
=
10
5
Define or explain the following terms:
1) Precision 2) Accuracy
3) Statistical independence 4) Type I error
5) Type II error 6) Expectation
7) Degree of freedom of a linear system
15
6
Given 20 independent measurements of a distance made with the same standard deviation of 0.033m and the calculated mean value µ= 37.615m, construct a 95% confidence interval for the population mean.
Percentiles of Standard Normal Distribution
α 0.002 0.003 0.004 0.005 0.01 0.025 0.05 Kα 2.88 2.75 2.65 2.58 2.33 1.96 1.64
where Kα is determined by the equation α= e dx 2
1 x /2
K
− 2
∫∞
α π .
10
7
A baseline of calibrated length (µ) 1153.00m is measured 5 times independently with the same precision. The sample mean ( x ) and sample standard deviation (s) are calculated from the measurements:
x =1153.39m s = 0.06m
Test at a 90% confidence level (α) if the measured distance is significantly different from the calibrated distance.
Percentiles of t distribution tα
Degree of
freedom 0.90
t t0.95 t0.975 t0.99
1 3.08 6.31 12.7 31.8
2 1.89 2.92 4.30 6.96
3 1.64 2.35 3.18 4.54
4 1.53 2.13 2.78 3.75
5 1.48 2.01 2.57 3.36
10
8
Given in the following is a diagram of a leveling network to be adjusted.
Table 1 contains all of the height difference measurements ∆hij necessary to carry out the adjustment. Assume that all observations have the same standard deviation
hij
σ∆ = 0.04m. P1 is a fixed point with known elevations of H1 = 101.00m. Perform least squares adjustment on the leveling network to compute:
a) the adjusted elevations of point P2, P3 and P4 and their variance- covariance matrix;
b) the observation residuals;
c) the adjusted height differences; and d) the a-posteriori variance factor σˆ0. Table 1
No. i j ∆hij(unit: metre)
1 P1 P3 6.15
2 P1 P4 12.54
3 P3 P4 6.46
4 P1 P2 1.11
5 P2 P4 11.55
6 P2 P3 5.10
25
Total Marks: 100 P1
P2
P3 P4
![The proof turns out to be a fairly straightforward extension of the proof of the main result in [4]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)